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Indium antimonide energy band parameters derived from diamagnetic exciton spectra
~
Solid State Communications, Vol.43,No.8, pp.613-617, 1 9 8 2 . Printed in Great Britain.
0038-1098/82/320613-05503.00/0 Pergamon Press Ltd.
INDI~ ~TI~IDE ENERGY BAND P A R A ~ T E R S DERIVED FRO~I DIA//AGNETIC EXCITON SPECTRA A1.L.Efros,
L.~.Kanskaya,
Ioffe Physicotechnical
S.I.Kokhanovskii
and R.~.Seysyan
Institute, Leningrad,
194021, USSR
(Received 19 April 1982 by V.M.Agranovich) The oscillatory absorption edge spectrum of thin InSb crystals in which the discrete structure of the Wannier-Mott exciton had been revealed at H=O for the first time has been studied at 1.8 K in magnetic fields of up to 80 kOe. The experimental data were analyze~ with a computer taking into account the exciton nature of the phenomenon and the effect on the Landau levels of the nonlocality of potential resulting from the electron-electron exchange interaction. A set of band parameters has been obtained providing a minimum discrepancy between the theoretical and experimental spectra by not more th~u 0.65 meV per point: Ep = 23.42 eV, ~ = -1~18, ~I " - 0 . 3 3 , ~36.41, ~ = 15.95, ~ - 16.99, ~ 15.22, q - 0.31 for ~ £ - 17.9. The reasons for and the extent of discrepancy between the InSb parameters derived by other authors from interband magnstoabsorption experiments are discussed.
I. ~he indium antimonide energy band parameters derived by Pigeon et al.1, 2 from oscillatory magnetoabsorption ( ~ ) experiments have been considered until recently as the most reliable. They were obtained, however, neglecting the exciton nature of the spectra. Recently new publications have appeared 3'4'5 on the determination of the InSb ~and parameters, including an attempt~ to take into consideration Coulomb interaction in an approximate way; the band parameter sets thus obtained turned out, however, to differ markedly not only from those of ref. 1'2 but from one another as wel~, which casts doubt on the accuracy of both "new" and "old" data on the band structure of InSb as well as on the validity of the method of calculation involved. A procedure for calculating the band parameters of diamond-lattice semiconductors in strong ~a6nuetic fields where the condition ~>> I is met ( ~ ; ~ q / 2 R ~ , where ~ is the sum of the electro~ and hole cyclotron frequencies, and R~ is the exciton ground state binding e n e ~ j ) has been proposed and 6 used in an analysis of CI,~ data for G a S ~ This opens up a possibility of discarding the very rough technique of calculation of ~he diamagnetic exciton binding energy R H suggested by Vrehen 7 for GaAs and used in ref. 3 613
The ~ . spectra corrected for the binding energy yielded ordinarily parameters approaching closely the most reliable data derived from cyclotron resonance 6. To ca~i~ out binding energy calculations by this technique even for relatively wlde-gap semiconductors, however, one had to take into account also the electron and light-hole nonparabolicity. As for the narrow-gap diamond-lattice semiconductors, one has to include in addition the nonlocality of potential originating from exchange interaction8. Neglect of this phenomenon may distort substantially the results of OMA data analysis. We have considered this to justify a study of the diamagnetic exciton spectra in InSb (as a development of the successful observation in InSb of the discrete nature of the Wannier-~ott exciton I0) focusing particular attention on an exact calculation of the binding energy of the diamagnetic exciton (DE) and a thorough qualitative and quantitative interpretation of the spectrum. 2. We have used in the experiments indium antimonide single c~,stals primarily of n-type with electron concentration n77 = 6x1013cm -3 and mobility ~ 7 ~ = 600,000 cmJV.s. The preparation of 6-8 ~m thick samples involved mechanical and chemical processing and a heat
INDIUM ANTIMONIDE ENERGY BAND PARAMETERS
614
treatment in hydrogen 11 • The samples together with a superconducting magnet coll capable of producing fields of up to 80 kOe were submerged in liquid helium maintained u ~ e r pumping. The sample w~s illuminated with monochromatic polarized light. A ~odel MDR-2 monochromator with a diffraction grating of 100 lines per ram, as well as most of the optical system, were kept in rough vacuum to exclude atmospheric absorption lines. A PbTe photoresistor cooled down to 77 K or a germanium bolometer maintained at 4.2 K served as detectors. A CaP o grating ~ s used to produce linear ~polarization, circular polarization being obtained by means of a NaCI l~resnel rhemb. 3. In the present communication we report on C~A measurements carried out on InSb in magnetic fields up to 80 kOe at T=1.8 K. Both the Coulomb interaction between the electron and the hole and the potential nomlocality effects are included in the analysis. The set of the ImSb energy bau~ parameters thus obtained will be compared with the other data available at present. Fig.1 displays the spectra measured by us at H=38.7 kOe. The theoretical spectra drawn below were calculated under the assumption of the OMA spectra originatizlg from transitions which satisfy the principal selection rules One Immediately sees that the experimental spectra fit well to the theoretical predictions without any arbitrary displacement or deformation. Such phenomena as band warping or inversion asymmetry do not practically affect the shape of the 0NA spectra in this material. As an exclusion, one may refer to the "extra" transitions b+(-1)aC(o) (the main transition being a-(1 )aC(o)) in the and
~ + spectrum, and a-(1)bC(0)
b+(-1)aC(2)
(the main transitions
are b+(-1)bc(O) and a-(1)aC(2), accordingly) in the ~ - s p e c t r u m with H II <111> which originate from the energy surface warping and produce strong maxima (Fig. I ) • The data derived from the ONA spectra are summarized in Table I. These data were obtained by fitting the theoretical O~L& spectlx~m to the maxima in the absorption coefficient of InSb observed in a magnetic field H=38.7 kOe for the ~ + and ~ - polarizations and for three orientations of the magnetic field, H II <1oo> , H I1<111) and H II <110~ . The total number of experimental points (maxima in the absorption coefficient) used in the least squares fitting was 125. Also presented are the data obtained in a similar experiment of Weiler. 3
Vol. 43, No. 8
In our analysis of the experiment on InSb we have attempted to est~m-te the effect of various model approximations on the shape of 0 ~ spectra in narrow-gap semiconductors. Therefore the Table may be divided into three parts, the first part illustrating the fitting of the theory to experiment without taking into account Coulomb interaction between the electron and the hole. The m~Timp in the absorption coefficient are assumed to be directly connected with electron transitions between T ~ d a u sub-bands of the valence and conduction bands. The position of the Landau levels was calculated by means of the equations of Pigeon and Brown I (PB) corrected and modified in ref 12. These equations take into accouat the nonparabolicity of the electron and light-hole spectra a ~ w e l l as deviations from Kane' s model Jassociated with the effect of higher bands. The second part of the Table lists the results of fitting of the theoretical OMA spectra considered as DE spectra. The DE binding energies were calculated by the expressions from ref. 6 taking account of the valence band degeneracy, as well as the nonparabolicity of the electron and light-hole spec8 tra. Finally, the third part of the Table contains the corrections to the Landau levels which cannot be obtained within the framework of the thecry assn~mlng only the local potential to act on the electrons 9'14 and the changes in the band parameters resulting from this effect. These phenomena can be significant only in narrow-gap semiconductors. For the bandgap energy used in the determination of the band parameters we took the value C g = 0.2368 eV derived from a direct observation of the discrete nature of the exciton in a zero magnetic field. Part I of the Table contains the results derived from our experimental data by the technique of ref. ~. In the least-squares fitting we used the following relati on,hip ~
~<--
32
~.~F~I +.~
+ ~-
(1)
We took q = 0.39 which is the value obtained in ref. 2. Our resu~t~ are seen to a~proach those of ref. ",~ (see columns 2 and 3). 9~11 agreement is seen only for the electron effective mass,~ m~ ffi 0.O145 m. As pointed out in ref. ~, this quantity is the most sensitive to the fitting procedure used. As seen from the second part of the the Table, inclusion of the exoiton nature of the O~h% spectra results in a substantial decrease of ~ . Reduced to one line, the RES deviation decreas-
615
INDIUM ANTIHONIDE ENERGY BAND PARAMETERS
Vol. 43, No. 8
(a) 2
I
~
I l
5
\
1
4, , cP "I~ I
~ll
<:P~ lcP~'
~'I'I '~
(b)
5
ix
I I
IJ
V
J 21
V6
3
fl
|
~J~'+m
If
2
3~
T,T
~ 2
5,
,o
|7~1 6
9~e ~
8
W
,vu
HV^
,~'
%,,,~2o2,~
~a4
"
13;,,,
"'a~:
1617
23
-',
'~
1819 2Or..
r.,.
"-
't,'t, ,P,,'r, 'PT,,, ,i,I, vT+,,PT .,,+,I' ,,,P| ,P'r l
(e)
I
2
'P,T 'PT, 'Pt, TTt 'PI ,<+I' d , ,T ,I' (f)
'-i 678
1 0.27
0.25
0.29
0.31
9 I0
0.33
II 12
14
035
0.37
0+39
0.41
0.43
0,45
E, eV
Fig. 1.
OD:J.
spectra in InSb obtained in the l~araday c o n f i g u r a t i o n
for three m a j o r c r y s t a l l o g r a p h i c
orientations
at H = 3 8 . 7 k0e,
1.8 K . 6
spectra:
~tspectra:
a - H
II
~ 111>
o-H
II <1oo>
d - H
II < 1 1 1 >
f-H
II < 1 0 0 >
, b - H
II d 1 1 0 >
e - H
II < 1 1 0 ~ caption
continued
INDIUM ANTIMONIDE ENERGY BAND PARAMETERS
6]6
Vol. 43, No. 8
Theoretical spectra were drawn taking into account the binding energy of the diamagnetic exciton ground state ( ~ = 0). Notations of the transitions : - a+(1) aC(l~1)
x - b+(1) bC(l!1)
O - a-(l) aC(l!1)
•-
b-(1) bC(l!1)
The ms~v~m~ corresponding to the excited states ( ~ = 1) of the diamagnetic exciton are denoted by an arrow. The additional absorption lines appearing only in the H
II ~111~
orientation
and originating in the warping-induced transitions are shov~ by a dashed arrow. The brace specifies the lines corresponding to the Larsen-Johnson polaron anomalies. Table I. Electron and hole energy band parameters for indium antimonide Excitons not .included
Para- a meter
I
Our data
ref.
Di~m,gnetic excitons (DE) included ref. 3
Our data Our data with coup- without ling (I) coupling
1
2
~S , eV
0,2355
0,2368
0,2329
0.2368
0.2368
0.2368
Ep, eV
21.9
22.48
23.5
23.44
23.43
23.42
F
0
-0.51
-1.0
-1.16
-1.17
-1.18
NI
0
-0.3
-0.22
-0.31
-0.33
m~/m
4
0
6
DE and nonlocality included, our data
7
0.0145
0.0145
0.0136
0.0139
0.0139
0.0139
-47.1
-48.5
-51.4
-51.5
-51.9
-51.9
32.5
34.6
37.1
36.11
36.29
36.41
14.3
15.3
16.5
15.93
15.97
15.95
15.4
16.35
17.7
16.96
17.01
16.99
13.4
14.31
15.6
14.87
15.19
15.22
1.5
3.02
3.4
3.11
3.31
3.44
-1.2
-0.54
-0.3
-0.57
-0.52
-0.54
~t
-0.1
0.51
0.9
0.46
0.52
0.51
~
-2.1
-1.51
-1.2
-I .63
~i
q ~
meV
a
0
0.39
2.7
1.1
0 4.2
-1.3
-1.26
0
0.32
0.31
0.68
0.66
0.65
The notations are the same as in refs.
I-3
Vol.
43, No. 8
INDIUM ANTIMONIDE
es almost by a factor 1.7 becoming 0.7 meV. Col.6 of Table I corresponds to fitting without invoking any coupling between the band parameters. The parameter ~ is also adjusted here to ensure the best fit. Coi.5 contains data obtained by using expression (I) at a fixed q=O. One sees ~ to be here larger than 8~ by 3 %. Col.4 contains the data derived from an CEA experiment taking into account the exciton effects 3. The values in ref. 3 are ~g= 0.2329 eV, which is stated to yield the smallest ~ and = 15.4. The reason for such a large discrepancy for the quantities gg and ~ and, consequently, for other parameters as well, is unclear to us. It is possibly due to the fact that the experimental technique of Weiler 3 results in a creation of stress in the crystal. Note that a value ~m= = 0.2367 eV close to our result is iiyen also in another publication 1 5 . The data listed in the third part of the Table were obtained after introducing corrections to the heavy-hole spectrum 16 which originates from the nonlocality of the potential. The results of the fitting performed without invoking any coupling between the band
ENERGY BAND PARAMETERS
parameters are presented in coi.7. Comparing these results with the data of Col.6 one readily sees that although the difference in the sum of the squares of deviations, 8~-8~, is not large (which is only n a ~ l considering the magnitude of the corrections obtained for the field H = = 38.7 kOe) taking into account the nonlocality of potential improves the agreement between the theoretical and experimental spectra• Our study shows that taking into account Coulomb interaction affects markedly the agreement between the theory and experiment by reducing the R~S deviation by almost a factor of 2, and including the nonlocality improves this agreement still more. Thus basing on our experimental data, one may suggest the following values of the band parameters which provide the best fit for the electronic processes involved: Ep = 23.42 eV, F = -1.18, N 4= -0.33, which corresponds to m~ = 0.0139 m and 8~ = -50.2, and ~ = 36.41, ~ = = 15.95, ~ = 16.99, k = 15.22, q = = 0.31. Acknowledgement - The authors wish to thank B.P.Zakharchenya for his stimulating interes~ in this work.
REFERENCES 1. 2. 3. 4.
PIDGEON C.R. & BRO°~ R.N., Phys.Rev., 146, 575 (1966). PIDGEON C.R. ~ GROVES S.H., Phys.Rev.,~___~6, 824 (1969). WEILER ~,I.H., J.~agn.Magn.~aterials, 11, ~31 (19.9). RANVALD R., TREBIN H.R., ROESSLER V . & POLI~%K F.~., Phye.Rev. B20, 701 (1979). . GRISAR R., Solid State Commun., 25, 1023 (1978). • GELMONT B.L., SEYSYAN R.P., E P R O ~ A 1 . L . & VARFOLOMEEV A.V., Fiz.Tekh.Polupr. 11, 1067 (1977). 7. V ~ H E N Q.E.F., J.Phys.Chem.Sollds, 29, 129 (1968). 8. VAR~OLO~,~V A.V., SEYSYAN R.P., E F R ~ A1.L., ZAKHARCHENYA B.P. & RYSKIN ~ ¥ a . Fiz.Tekh.Polupr., 11, 2301 (1977). 9. GELMONT B.L. & E P R O S A 1 . L . , Fiz.Tekh.Polupr. 13, 248 (1979). 10. KANSKAYA L.E., K O K H ~ O V S K I I S.I. & SEYSYAN R.~., Fiz.Tekh.Polupr., 13, 2424 (1979). 11. ABDULAEV E.A., AGEKYAN V.T. & SEYSYA~ R.P., Fiz.Tekh.Polupr., 7, 2217 (1973). 12. AGGARWAL R.J., Phys.Rev. B2, 446 (1970). 13. Y~AME E.O., J.Phys.Chem.Solids, 1, 249 (1957). 14. GEI~ONT B.L., Fiz.Tekh.Polupr.,--~, 1912 (1975). 15. JOHNSON E.J., Phys.Rev.Lett., 19, 352 (1967). 16. GEI~IONT B.L., SEYSYAN R.P. & EFROS A1.L., Fiz.Tekh.Polupr., 15, 776 (1982).
617
Solid State Communications, Vol.43,No.8, pp.613-617, 1 9 8 2 . Printed in Great Britain.
0038-1098/82/320613-05503.00/0 Pergamon Press Ltd.
INDI~ ~TI~IDE ENERGY BAND P A R A ~ T E R S DERIVED FRO~I DIA//AGNETIC EXCITON SPECTRA A1.L.Efros,
L.~.Kanskaya,
Ioffe Physicotechnical
S.I.Kokhanovskii
and R.~.Seysyan
Institute, Leningrad,
194021, USSR
(Received 19 April 1982 by V.M.Agranovich) The oscillatory absorption edge spectrum of thin InSb crystals in which the discrete structure of the Wannier-Mott exciton had been revealed at H=O for the first time has been studied at 1.8 K in magnetic fields of up to 80 kOe. The experimental data were analyze~ with a computer taking into account the exciton nature of the phenomenon and the effect on the Landau levels of the nonlocality of potential resulting from the electron-electron exchange interaction. A set of band parameters has been obtained providing a minimum discrepancy between the theoretical and experimental spectra by not more th~u 0.65 meV per point: Ep = 23.42 eV, ~ = -1~18, ~I " - 0 . 3 3 , ~36.41, ~ = 15.95, ~ - 16.99, ~ 15.22, q - 0.31 for ~ £ - 17.9. The reasons for and the extent of discrepancy between the InSb parameters derived by other authors from interband magnstoabsorption experiments are discussed.
I. ~he indium antimonide energy band parameters derived by Pigeon et al.1, 2 from oscillatory magnetoabsorption ( ~ ) experiments have been considered until recently as the most reliable. They were obtained, however, neglecting the exciton nature of the spectra. Recently new publications have appeared 3'4'5 on the determination of the InSb ~and parameters, including an attempt~ to take into consideration Coulomb interaction in an approximate way; the band parameter sets thus obtained turned out, however, to differ markedly not only from those of ref. 1'2 but from one another as wel~, which casts doubt on the accuracy of both "new" and "old" data on the band structure of InSb as well as on the validity of the method of calculation involved. A procedure for calculating the band parameters of diamond-lattice semiconductors in strong ~a6nuetic fields where the condition ~>> I is met ( ~ ; ~ q / 2 R ~ , where ~ is the sum of the electro~ and hole cyclotron frequencies, and R~ is the exciton ground state binding e n e ~ j ) has been proposed and 6 used in an analysis of CI,~ data for G a S ~ This opens up a possibility of discarding the very rough technique of calculation of ~he diamagnetic exciton binding energy R H suggested by Vrehen 7 for GaAs and used in ref. 3 613
The ~ . spectra corrected for the binding energy yielded ordinarily parameters approaching closely the most reliable data derived from cyclotron resonance 6. To ca~i~ out binding energy calculations by this technique even for relatively wlde-gap semiconductors, however, one had to take into account also the electron and light-hole nonparabolicity. As for the narrow-gap diamond-lattice semiconductors, one has to include in addition the nonlocality of potential originating from exchange interaction8. Neglect of this phenomenon may distort substantially the results of OMA data analysis. We have considered this to justify a study of the diamagnetic exciton spectra in InSb (as a development of the successful observation in InSb of the discrete nature of the Wannier-~ott exciton I0) focusing particular attention on an exact calculation of the binding energy of the diamagnetic exciton (DE) and a thorough qualitative and quantitative interpretation of the spectrum. 2. We have used in the experiments indium antimonide single c~,stals primarily of n-type with electron concentration n77 = 6x1013cm -3 and mobility ~ 7 ~ = 600,000 cmJV.s. The preparation of 6-8 ~m thick samples involved mechanical and chemical processing and a heat
INDIUM ANTIMONIDE ENERGY BAND PARAMETERS
614
treatment in hydrogen 11 • The samples together with a superconducting magnet coll capable of producing fields of up to 80 kOe were submerged in liquid helium maintained u ~ e r pumping. The sample w~s illuminated with monochromatic polarized light. A ~odel MDR-2 monochromator with a diffraction grating of 100 lines per ram, as well as most of the optical system, were kept in rough vacuum to exclude atmospheric absorption lines. A PbTe photoresistor cooled down to 77 K or a germanium bolometer maintained at 4.2 K served as detectors. A CaP o grating ~ s used to produce linear ~polarization, circular polarization being obtained by means of a NaCI l~resnel rhemb. 3. In the present communication we report on C~A measurements carried out on InSb in magnetic fields up to 80 kOe at T=1.8 K. Both the Coulomb interaction between the electron and the hole and the potential nomlocality effects are included in the analysis. The set of the ImSb energy bau~ parameters thus obtained will be compared with the other data available at present. Fig.1 displays the spectra measured by us at H=38.7 kOe. The theoretical spectra drawn below were calculated under the assumption of the OMA spectra originatizlg from transitions which satisfy the principal selection rules One Immediately sees that the experimental spectra fit well to the theoretical predictions without any arbitrary displacement or deformation. Such phenomena as band warping or inversion asymmetry do not practically affect the shape of the 0NA spectra in this material. As an exclusion, one may refer to the "extra" transitions b+(-1)aC(o) (the main transition being a-(1 )aC(o)) in the and
~ + spectrum, and a-(1)bC(0)
b+(-1)aC(2)
(the main transitions
are b+(-1)bc(O) and a-(1)aC(2), accordingly) in the ~ - s p e c t r u m with H II <111> which originate from the energy surface warping and produce strong maxima (Fig. I ) • The data derived from the ONA spectra are summarized in Table I. These data were obtained by fitting the theoretical O~L& spectlx~m to the maxima in the absorption coefficient of InSb observed in a magnetic field H=38.7 kOe for the ~ + and ~ - polarizations and for three orientations of the magnetic field, H II <1oo> , H I1<111) and H II <110~ . The total number of experimental points (maxima in the absorption coefficient) used in the least squares fitting was 125. Also presented are the data obtained in a similar experiment of Weiler. 3
Vol. 43, No. 8
In our analysis of the experiment on InSb we have attempted to est~m-te the effect of various model approximations on the shape of 0 ~ spectra in narrow-gap semiconductors. Therefore the Table may be divided into three parts, the first part illustrating the fitting of the theory to experiment without taking into account Coulomb interaction between the electron and the hole. The m~Timp in the absorption coefficient are assumed to be directly connected with electron transitions between T ~ d a u sub-bands of the valence and conduction bands. The position of the Landau levels was calculated by means of the equations of Pigeon and Brown I (PB) corrected and modified in ref 12. These equations take into accouat the nonparabolicity of the electron and light-hole spectra a ~ w e l l as deviations from Kane' s model Jassociated with the effect of higher bands. The second part of the Table lists the results of fitting of the theoretical OMA spectra considered as DE spectra. The DE binding energies were calculated by the expressions from ref. 6 taking account of the valence band degeneracy, as well as the nonparabolicity of the electron and light-hole spec8 tra. Finally, the third part of the Table contains the corrections to the Landau levels which cannot be obtained within the framework of the thecry assn~mlng only the local potential to act on the electrons 9'14 and the changes in the band parameters resulting from this effect. These phenomena can be significant only in narrow-gap semiconductors. For the bandgap energy used in the determination of the band parameters we took the value C g = 0.2368 eV derived from a direct observation of the discrete nature of the exciton in a zero magnetic field. Part I of the Table contains the results derived from our experimental data by the technique of ref. ~. In the least-squares fitting we used the following relati on,hip ~
~<--
32
~.~F~I +.~
+ ~-
(1)
We took q = 0.39 which is the value obtained in ref. 2. Our resu~t~ are seen to a~proach those of ref. ",~ (see columns 2 and 3). 9~11 agreement is seen only for the electron effective mass,~ m~ ffi 0.O145 m. As pointed out in ref. ~, this quantity is the most sensitive to the fitting procedure used. As seen from the second part of the the Table, inclusion of the exoiton nature of the O~h% spectra results in a substantial decrease of ~ . Reduced to one line, the RES deviation decreas-
615
INDIUM ANTIHONIDE ENERGY BAND PARAMETERS
Vol. 43, No. 8
(a) 2
I
~
I l
5
\
1
4, , cP "I~ I
~ll
<:P~ lcP~'
~'I'I '~
(b)
5
ix
I I
IJ
V
J 21
V6
3
fl
|
~J~'+m
If
2
3~
T,T
~ 2
5,
,o
|7~1 6
9~e ~
8
W
,vu
HV^
,~'
%,,,~2o2,~
~a4
"
13;,,,
"'a~:
1617
23
-',
'~
1819 2Or..
r.,.
"-
't,'t, ,P,,'r, 'PT,,, ,i,I, vT+,,PT .,,+,I' ,,,P| ,P'r l
(e)
I
2
'P,T 'PT, 'Pt, TTt 'PI ,<+I' d , ,T ,I' (f)
'-i 678
1 0.27
0.25
0.29
0.31
9 I0
0.33
II 12
14
035
0.37
0+39
0.41
0.43
0,45
E, eV
Fig. 1.
OD:J.
spectra in InSb obtained in the l~araday c o n f i g u r a t i o n
for three m a j o r c r y s t a l l o g r a p h i c
orientations
at H = 3 8 . 7 k0e,
1.8 K . 6
spectra:
~tspectra:
a - H
II
~ 111>
o-H
II <1oo>
d - H
II < 1 1 1 >
f-H
II < 1 0 0 >
, b - H
II d 1 1 0 >
e - H
II < 1 1 0 ~ caption
continued
INDIUM ANTIMONIDE ENERGY BAND PARAMETERS
6]6
Vol. 43, No. 8
Theoretical spectra were drawn taking into account the binding energy of the diamagnetic exciton ground state ( ~ = 0). Notations of the transitions : - a+(1) aC(l~1)
x - b+(1) bC(l!1)
O - a-(l) aC(l!1)
•-
b-(1) bC(l!1)
The ms~v~m~ corresponding to the excited states ( ~ = 1) of the diamagnetic exciton are denoted by an arrow. The additional absorption lines appearing only in the H
II ~111~
orientation
and originating in the warping-induced transitions are shov~ by a dashed arrow. The brace specifies the lines corresponding to the Larsen-Johnson polaron anomalies. Table I. Electron and hole energy band parameters for indium antimonide Excitons not .included
Para- a meter
I
Our data
ref.
Di~m,gnetic excitons (DE) included ref. 3
Our data Our data with coup- without ling (I) coupling
1
2
~S , eV
0,2355
0,2368
0,2329
0.2368
0.2368
0.2368
Ep, eV
21.9
22.48
23.5
23.44
23.43
23.42
F
0
-0.51
-1.0
-1.16
-1.17
-1.18
NI
0
-0.3
-0.22
-0.31
-0.33
m~/m
4
0
6
DE and nonlocality included, our data
7
0.0145
0.0145
0.0136
0.0139
0.0139
0.0139
-47.1
-48.5
-51.4
-51.5
-51.9
-51.9
32.5
34.6
37.1
36.11
36.29
36.41
14.3
15.3
16.5
15.93
15.97
15.95
15.4
16.35
17.7
16.96
17.01
16.99
13.4
14.31
15.6
14.87
15.19
15.22
1.5
3.02
3.4
3.11
3.31
3.44
-1.2
-0.54
-0.3
-0.57
-0.52
-0.54
~t
-0.1
0.51
0.9
0.46
0.52
0.51
~
-2.1
-1.51
-1.2
-I .63
~i
q ~
meV
a
0
0.39
2.7
1.1
0 4.2
-1.3
-1.26
0
0.32
0.31
0.68
0.66
0.65
The notations are the same as in refs.
I-3
Vol.
43, No. 8
INDIUM ANTIMONIDE
es almost by a factor 1.7 becoming 0.7 meV. Col.6 of Table I corresponds to fitting without invoking any coupling between the band parameters. The parameter ~ is also adjusted here to ensure the best fit. Coi.5 contains data obtained by using expression (I) at a fixed q=O. One sees ~ to be here larger than 8~ by 3 %. Col.4 contains the data derived from an CEA experiment taking into account the exciton effects 3. The values in ref. 3 are ~g= 0.2329 eV, which is stated to yield the smallest ~ and = 15.4. The reason for such a large discrepancy for the quantities gg and ~ and, consequently, for other parameters as well, is unclear to us. It is possibly due to the fact that the experimental technique of Weiler 3 results in a creation of stress in the crystal. Note that a value ~m= = 0.2367 eV close to our result is iiyen also in another publication 1 5 . The data listed in the third part of the Table were obtained after introducing corrections to the heavy-hole spectrum 16 which originates from the nonlocality of the potential. The results of the fitting performed without invoking any coupling between the band
ENERGY BAND PARAMETERS
parameters are presented in coi.7. Comparing these results with the data of Col.6 one readily sees that although the difference in the sum of the squares of deviations, 8~-8~, is not large (which is only n a ~ l considering the magnitude of the corrections obtained for the field H = = 38.7 kOe) taking into account the nonlocality of potential improves the agreement between the theoretical and experimental spectra• Our study shows that taking into account Coulomb interaction affects markedly the agreement between the theory and experiment by reducing the R~S deviation by almost a factor of 2, and including the nonlocality improves this agreement still more. Thus basing on our experimental data, one may suggest the following values of the band parameters which provide the best fit for the electronic processes involved: Ep = 23.42 eV, F = -1.18, N 4= -0.33, which corresponds to m~ = 0.0139 m and 8~ = -50.2, and ~ = 36.41, ~ = = 15.95, ~ = 16.99, k = 15.22, q = = 0.31. Acknowledgement - The authors wish to thank B.P.Zakharchenya for his stimulating interes~ in this work.
REFERENCES 1. 2. 3. 4.
PIDGEON C.R. & BRO°~ R.N., Phys.Rev., 146, 575 (1966). PIDGEON C.R. ~ GROVES S.H., Phys.Rev.,~___~6, 824 (1969). WEILER ~,I.H., J.~agn.Magn.~aterials, 11, ~31 (19.9). RANVALD R., TREBIN H.R., ROESSLER V . & POLI~%K F.~., Phye.Rev. B20, 701 (1979). . GRISAR R., Solid State Commun., 25, 1023 (1978). • GELMONT B.L., SEYSYAN R.P., E P R O ~ A 1 . L . & VARFOLOMEEV A.V., Fiz.Tekh.Polupr. 11, 1067 (1977). 7. V ~ H E N Q.E.F., J.Phys.Chem.Sollds, 29, 129 (1968). 8. VAR~OLO~,~V A.V., SEYSYAN R.P., E F R ~ A1.L., ZAKHARCHENYA B.P. & RYSKIN ~ ¥ a . Fiz.Tekh.Polupr., 11, 2301 (1977). 9. GELMONT B.L. & E P R O S A 1 . L . , Fiz.Tekh.Polupr. 13, 248 (1979). 10. KANSKAYA L.E., K O K H ~ O V S K I I S.I. & SEYSYAN R.~., Fiz.Tekh.Polupr., 13, 2424 (1979). 11. ABDULAEV E.A., AGEKYAN V.T. & SEYSYA~ R.P., Fiz.Tekh.Polupr., 7, 2217 (1973). 12. AGGARWAL R.J., Phys.Rev. B2, 446 (1970). 13. Y~AME E.O., J.Phys.Chem.Solids, 1, 249 (1957). 14. GEI~ONT B.L., Fiz.Tekh.Polupr.,--~, 1912 (1975). 15. JOHNSON E.J., Phys.Rev.Lett., 19, 352 (1967). 16. GEI~IONT B.L., SEYSYAN R.P. & EFROS A1.L., Fiz.Tekh.Polupr., 15, 776 (1982).
617